Intertwining Multiresolution Analyses and the Construction of Piecewise-Polynomial Wavelets

Let (Vₚ ) be a local multiresolution analysis of L² ( R) of multiplicity r ≥ 1, i.e., V₀ is generated by r compactly supported scaling functions. If the scaling functions generate an orthogonal basis of V₀, then (Vₚ ) is called an orthogonal multiresolution analysis. We prove that there exists an orthogonal local multiresolution analysis (V'ₚ ) of multiplicity $r'$ such that \[V_q ⊂ V'_0 ⊂ Vq + n \] for some integers q ≥ 0, n ≥ 1, and $r' > 1$. In particular, this shows that compactly supported orthogonal polynomial spline wavelets and scaling functions (of multiplicity $r' > 1$) of arbitrary regularity exist, and we give several such examples.